Multi-scale computational method for elastic bodies with global and local heterogeneity

A multi-scale computational method using the homogenization theory and the finite element mesh superposition technique is presented for the stress analysis of composite materials and structures from both micro- and macroscopic standpoints. The proposed method is based on the continuum mechanics, and the micro–macro coupling effects are considered for a variety of composites with very complex microstructures. To bridge the gap of the length scale between the microscale and the macroscale, the homogenized material model is basically used. The classical homogenized model can be applied to the case that the microstructures are periodically arrayed in the structure and that the macroscopic strain field is uniform within the microscopic unit cell domain. When these two conditions are satisfied, the homogenization theory provides the most reliable homogenized properties rigorously to the continuum mechanics. This theory can also calculate the microscopic stresses as well as the macroscopic stresses, which is the most attractive advantage of this theory over other homogenizing techniques such as the rule of mixture. The most notable feature of this paper is to utilize the finite element mesh superposition technique along with the homogenization theory in order to analyze cases where non-periodic local heterogeneity exists and the macroscopic field is non-uniform. The accuracy of the analysis using the finite element mesh superposition technique is verified through a simple example. Then, two numerical examples of knitted fabric composite materials and particulate reinforced composite material are shown. In the latter example, a shell-solid connection is also adopted for the cost-effective multi-scale modeling and analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Characterization of heterogeneous materials under shear loading at finite strain

This article presents a homogenization procedure to predict the effective shear response of heterogeneous materials at large deformation. Assuming local periodicity, heterogeneous microstructure is identified by a representative volume element that is subjected to an equivalent macroscopic deformation field. The energy balance and periodicity conditions are considered to relate macro and micro-stress fields. Based on the symmetrical planes of the microstructure and local periodicity, it is shown that the analysis of one-quarter of three-dimensional representative volume element is enough to evaluate the effective shear response at finite deformation. A computational method is subsequently developed to obtain the shear response of heterogeneous microstructures. The homogenization procedure is implemented to evaluate shear response of two specific heterogeneous materials, elastomeric composite and reinforced viscoelastic fluid. The performance is successfully verified by comparison of the deformation in the macroscopic level to the response of a homogenized cell.     

A numerical method for homogenization in non-linear elasticity

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.     

Thermal contact conductance characterization via computational contact homogenization: A finite deformation theory framework

In order to predict the macroscopic thermal response of contact interfaces between rough surface topographies, a computational contact homogenization technique is developed at the finite deformation regime. The overall homogenization framework transfers macroscopic contact variables, such as surfacial stretch, pressure and heat flux, as boundary conditions on a test sample within a micromechanical interface testing procedure. An analysis of the thermal dissipation within the test sample reveals a thermodynamically consistent identification for the macroscopic thermal contact conductance parameter that enables the solution of a homogenized thermomechanical contact boundary value problem based on standard computational approaches. The homogenized contact response effectively predicts a temperature jump across the macroscale contact interface. The strong dependence of this homogenized response on macroscale solution variables of interest is demonstrated via representative three‐dimensional numerical investigations. The proposed contact homogenization framework is suitable for the analysis of similar energy transport phenomena across heterogeneous contact interfaces where the investigation of the sources for energy dissipation is of concern. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite Element Analysis and Experimental Measurement of Deformation Behavior for NiTi Plate With Stress Concentration Part Under Uniaxial Tensile Loading

Abstract: The aim of this study is to verify the effectiveness of ordinary phenomenological constitutive relation of NiTi shape memory alloy under mechanical loading at a constant temperature, sufficiently. First, finite element analysis is performed by using ordinary phenomenological constitutive relation for rectangular plate with double notch under tensile loading at a constant temperature. Next, uniaxial tensile loading is carried out for 50.5Ni49.5Ti rectangular plate with double notch. At the same time, macroscopic stress–strain curve and local strain distribution are measured by using in‐house measurement system on the basis of digital image correlation. As a result, it is found that the stress–strain curve obtained from finite element analysis is much different from those obtained experimental measurement, especially during stress‐induced martensite transformation. The result can be derived from the phenomena of local strain band behavior arising in NiTi under mechanical loading. The phenomenological constitutive model used in present finite element analysis is constructed under assumptions that the material has isotropic characteristics and shows homogeneous deformation. However, this experimental result suggests that the material itself has anisotropy microscopically. Furthermore, material shows unique inhomogeneous deformation. Also, there is possibility that these anisotropic characteristic and inhomogeneous deformation behaviour may derive from its microstructure. In future, to sufficiently describe the macroscopic stress–strain curve of NiTi we should take into consideration the material microstructure.     

Multiscale design of three-dimensional nonlinear composites using an interface-enriched generalized finite element method

A computational framework is developed to model and optimize the nonlinear multiscale response of three-dimensional particulate composites using an interface-enriched generalized finite element method. The material nonlinearities are associated with interfacial debonding of inclusions from a surrounding matrix which is modeled using C⁻¹ continuous enrichment functions and a cohesive failure model. Analytic material and shape sensitivities of the homogenized constitutive response are derived and used to drive a nonlinear inverse homogenization problem using gradient-based optimization methods. Spherical and ellipsoidal particulate microstructures are designed to match a component of the homogenized stress-strain response to a desired constructed macroscopic stress-strain behavior.     

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C¹ approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.     

Computational multiscale modelling of heterogeneous material layers

A computational homogenization procedure for a material layer that possesses an underlying heterogeneous microstructure is introduced within the framework of finite deformations. The macroscopic material properties of the material layer are obtained from multiscale considerations. At the macro level, the layer is resolved as a cohesive interface situated within a continuum, and its underlying microstructure along the interface is treated as a continuous representative volume element of given height. The scales are linked via homogenization with customized hybrid boundary conditions on this representative volume element, which account for the deformation modes along the interface. A nested numerical solution scheme is adopted to link the macro and micro scales. Numerical examples successfully display the capability of the proposed approach to solve macroscopic boundary value problems with an evaluation of the constitutive properties of the material layer based on its micro-constitution.     

Three-dimensional computational micro-mechanical model for woven fabric composites

This work presents a computational material model for plain-woven fabric composite for use in finite element analysis. The material model utilizes the micro-mechanical approach and the homogenization technique. The micro-mechanical model consists of four sub-cells, however, because of the existing anti-symmetry only two sub-cells have to be homogenized for prediction of the elastic material properties. This makes the model computationally very efficient and suitable for large-scale finite element analysis. The model allows the warp and fill yarns not to be orthogonal in the plane of the composite ply. This gives the opportunity to model complex-shaped composite structures with different braid angles. General homogenization procedure is employed with two levels of property homogenization. The model is programmed in MATLAB software and the predicted material properties of different composite materials are compared and presented. The material model shows good capability to predict elastic material properties of composites and very good computational efficiency.     

Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum

In this paper the macroscopic creep and relaxation functions of a heterogeneous viscoelastic porous medium are derived by using Mori-Tanaka homogenization scheme. Analytical and semi-analytical solutions can then be determined with a parametric number of heterogeneous phases embedded in a viscoelastic matrix whose behavior is described with a parametric number of analogical units. Under some simplifying assumptions, a solution strategy is presented in order to make explicit how the microscopic retardation and relaxation times of the viscoelastic matrix control the distribution of the retardation and relaxation times of the homogenized medium.     

Micro-mechanical approach and algorithm for the study of damage appearance in elastomer composites

This paper aims with the development of a numerical non-incremental algorithm well suited for the homogenization of non-linearly elastic composites. On the one hand it allows to obtain the macroscopic behavior of such composites, without any numerical overcosts and on the other hand to estimate with a good accuracy the local forces sustained by the composite constituents. This latter property is very helpful in a further step when dealing with the simulation of damage evolution in structures of industrial relevance. In order to illustrate these homogenization and fast solution algorithm, they are both applied to the case of unidirectional composites. Thanks to the use of the finite element technique, the homogenized responses corresponding to some macroscopic loadings are first computed. Next, the analysis of three kinds of damage, which may be expected in such composites, is carried out. Finally the numerical performances of the proposed algorithm are presented and compared to these induced by an incremental algorithm.     

Two-scale analysis for deformation-induced anisotropy of polycrystalline metals

The anisotropic macroscopic mechanical behavior of polycrystalline metals is characterized by incorporating the microscopic constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory, which enables us to derive both micro- and macro-scale governing equations. The two-scale simulations are conducted to evaluate the macroscopic anisotropy induced by microscopic plastic deformation histories of the polycrystalline aggregate. In the simulations, the representative volume element (RVE) composed of several crystal grains is uniformly loaded in one direction, unloaded to macroscopically zero stress in a certain stage of deformation and then re-loaded in the different directions. The last re-loading calculations provide different macroscopic responses of the RVE, which can be the appearance of material anisotropy. We then try to examine the effects of the intergranular and intragranular behaviors on the anisotropy by means of various illustrations of microscopic plastic deformation process without referring to the change of crystallographic orientations.     

A microscopic nonlinear programming approach to shakedown analysis of cohesive–frictional composites

A microscopic approach together with nonlinear programming technique and finite element method is developed for shakedown analysis of a composite which has cohesive–frictional constituents. The macroscopic shakedown limit of a composite subject to cyclic loading is calculated in a direct way and the macro–micro relation is quantitatively evaluated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure – Representative Volume Element (RVE) chosen from a periodic heterogeneous material. Pressure-dependence and non-associated plastic flow of cohesive–frictional constituent materials are formulated into shakedown analysis. Based on the mathematical programming technique and the finite element method, the numerical micro-shakedown model is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized Lagrangian-penalty iterative algorithm. The proposed approach provides a direct approach for determining the reduced macroscopic strength domain of heterogeneous or composite materials due to cyclic loading. Meanwhile, it can capture different plastic behaviors of materials and therefore the developed method has a wide applicability.     

Limit analysis of composite materials with anisotropic microstructures: A homogenization approach

A nonlinear mathematical programming approach together with the finite element method and homogenization technique is developed to implement kinematic limit analysis for a microstructure and the macroscopic strength of a composite with anisotropic constituents can be directly calculated. By means of the homogenization theory, the classical kinematic theorem of limit analysis is generalized to incorporate the microstructure - Representative Volume Element (RVE) chosen from a periodic composite/heterogeneous material. Then, using an associated plastic flow rule, a general yield function is directly introduced into limit analysis and a purely-kinematic formulation is obtained. Based on the mathematical programming technique, the finite element model of microstructure is finally formulated as a nonlinear programming problem subject to only one equality constraint, which is solved by a direct iterative algorithm. The calculation is entirely based on a purely-kinematical velocity field without calculation of stress fields. Meanwhile, only one equality constraint is introduced into the nonlinear programming problem. So the computational cost is very modest. Both anisotropy and pressure-dependence of material yielding behavior are considered in the general form of kinematic limit analysis. The developed method provides a direct approach for determining the macroscopic strength domain of anisotropic composites and can serve as a powerful tool for microstructure design of composites.     

Multiscale analysis of laminated plates with integrated piezoelectric fiber composite actuators

This study is concerned with the detailed analysis of fiber-reinforced composite plates with integrated piezoceramic fiber composite actuators. A multiscale framework based on the asymptotic expansion homogenization method is used to couple the microscale and macroscale field variables. The microscale fluctuations in the mechanical displacement and electric potential are related to the macroscale deformation and electric fields through 36 distinct characteristic functions. The local mechanical and charge equilibrium equations yield a system of partial differential equations for the characteristic functions that are solved using the finite element method. The homogenized electroelastic properties of a representative material element are computed using the characteristic functions and the material properties of the fiber and matrix. The three-dimensional macroscopic equilibrium equations for a laminated piezoelectric plate are solved analytically using the Eshelby-Stroh formalism. The formulation admits different boundary conditions at the edges and is applicable to thick and thin laminated plates. The microscale stresses and electric displacement in the fibers and matrix are computed from the macroscale fields through interscale transfer operators. The multiscale analysis procedure is illustrated using two model problems. In the first model problem, a simply-supported sandwich plate consisting of a piezoceramic fiber composite shear actuator embedded between two graphite/polymer layers is studied. The second model problem concerns a cantilever graphite/polymer substrate with segmented piezoceramic fiber composite extension actuators attached to its top and bottom surfaces. Results are presented for the homogenized material properties, macroscale deformation, macroscale average stresses and microscale stress distributions.     

Multiscale prediction of mechanical behavior of ferrite–pearlite steel with numerical material testing

Multiscale mechanical behaviors of ferrite–pearlite steel were predicted using numerical material testing (NMT) based on the finite element method. The microstructure of ferrite–pearlite steel is regarded as a two‐component aggregate of ferrite crystal grains and pearlite colonies. In NMT, the macroscopic stress–strain curve and the deformation state of the microstructure were examined by means of a two‐scale finite element analysis method based on the framework of the mathematical homogenization theory. The microstructure of ferrite–pearlite steel was modeled with finite elements, and constitutive models for ferrite crystal grains and pearlite colonies were prepared to describe their anisotropic mechanical behavior at the microscale level. While the anisotropic linear elasticity and the single crystal plasticity based on representative characteristic length have been employed for the ferrite crystal grains, the constitutive model of a pearlite colony was newly developed in this study. For that reason, the constitutive behavior of the pearlite colony was investigated using NMT on a smaller scale than the scale of the ferrite–pearlite microstructure, with the microstructure of the pearlite colony modeled as a lamellar structure of ferrite and cementite phases with finite elements. On the basis of the numerical results, the anisotropic constitutive model of the pearlite colony was formulated based on the normal vector of the lamella. The components of the anisotropic elasticity were estimated with NMT based on the finite element method, where the elasticity of the cementite phase was numerically evaluated with a first‐principles calculation. Also, an anisotropic plastic constitutive model for the pearlite colony was formulated with two‐surface plasticity consisting of yield functions for the interlamellar shear mode and yielding of the overall lamellar structure. After addressing the microscopic modeling of ferrite–pearlite steel, NMT was performed with the finite element models of the ferrite–pearlite microstructure and with the microscopic constitutive models for each of the components. Finally, the results were compared with the corresponding experimental results on both the macroscopic response and the microscopic deformation state to ascertain the validity of the numerical modeling. Copyright © 2011 John Wiley & Sons, Ltd.